Wild Goose Chase

Occasionally, an individual (usually the hero) when being tracked, will discover the Tracking Device. One's first instinct may be to destroy the device, but why deprive oneself of the opportunity to mess with your enemies? So the hero promptly attaches the device to something else that (preferably) moves a lot and away from himself, causing the trackers to follow it all over the place before they finally discover that they've been duped.

Examples of Wild Goose Chase include:

Anime and Manga

In Fullmetal Alchemist, Ran Fan pulls off a much more primitive version of this by cutting off her own injured arm and tying it to a stray dog so that the Fuhrer, who is following the blood trail, is thrown off.
A variation of this happens in Ai no Kusabi novel and anime. The tracer is removed from the person being tracked so the pursuer is Lured into a Trap at another location.

Film

In Total Recall, the hero removes a tracking device from (apparently) his sinus cavity, shoves it in a bit of food, and tosses it to some nearby rats.
In another Philip K. Dick movie (can't remember title) the protagonist plays silly buggers with his pursuers by removing his tracking device and slipping into the pocket of the guy leading the hunt, leading to a lot of reports to the effect of "he just passed your position" before he figures it out.
Happens in Short Circuit when Number 5 notices his tracking device and throws it into the pick up truck of an elderly couple. The soldiers in pursuit of him find the couple instead.
Happens in Con Air when the convicts remove the plane's tracking device and places it on a scenic tours flight.
In The Siege Denzel Washington's character uses this without a tracking device as the US Army is tracking him. He has one of his agents dress like him and run in the wrong direction to divert attention away from his vehicle while he goes off to complete his original objective.

Literature

In the Dresden Files' Small Favors, Harry does a magical version of this by suspending the device over a large map of the city and enchanting it to make it believe that the map is the city. Then he attaches a bag of catnip to it and sets Mister on it. It proves very effective.
In The Da Vinci Code, Langdon attaches a bug to a bar of soap and throws it into the back of a garbage truck to distract the French police and buy more time in finding clues in the Louvre. Notable in that the Louvre, like most public buildings, doesn't have bar soap.

Live Action TV

In Buffy the Vampire Slayer, Xander does this inadvertently when Spike is tagged by the Initiative. After they remove the tracker, Xander flushes it, baffling the trackers.
In the Tek War TV show, federal agents find a tracking program that they believe is following the man they're looking for. It's actually tracking the exact location of a randomly selected pizza delivery guy, for the purpose of sending the Feds on one of these.
A variant is used in the Monk episode "Mr Monk and the Marathon Man." The eponymous Marathon Man commited the crime even though he had a chip in his shoe tracking his marathon time - because his chip wasn't in his shoe, it was in a camera trolly which was filming a famous marathoner as he ran his last race.

Western Animation

Once on The Simpsons, Marge uses the GPS chip on Bart's cell phone to track him down and keep him from mischief. Bart finds out and ties the chip to a bird, who ends up taking the family all the way to Machu Picchu.
Danger Mouse: "The Wild, Wild Goose Chase" is just that. DM discovers a device in a holed-out book that responds to Baron Greenback's voice and will tell where his next hideout is after telling where his present coordinates are. DM impersonates Greenback's voice with spooky perfection to learn the whereabouts of his next hideout, but each destination turns up a dead end. Concludes with an uncharacteristic Team Rocket Wins moment.